A STRATEGIC MARKET GAME
WITH ACTIVE BANKRUPTCY
J. GEANAKOPLOS, I. KARATZAS, M. SHUBIK and W. SUDDERTH∗
June 2, 2007
Abstract
We construct stationary Markov equilibria for an economy with fiat money, one nondurable commodity, countably-many time periods, and a continuum of agents. The total
production of commodity remains constant, but individual agents’ endowments fluctuate in
a random fashion from period to period. In order to hedge against these random fluctuations,
agents find it useful to hold fiat money which they can borrow or deposit at appropriate rates
of interest; such activity may take place either at a central bank (which fixes interest rates judiciously) or through a money-market (in which interest rates are determined endogenously).
We carry out an equilibrium analysis, based on a careful study of Dynamic Programming
equations and on properties of the invariant measures for associated optimally-controlled
Markov chains. This analysis yields the stationary distribution of wealth across agents, as
well as the stationary price (for the commodity) and interest rates (for the borrowing and
lending of fiat money).
A distinctive feature of our analysis is the incorporation of bankruptcy, both as a real
possibility in an individual agent’s optimization problem, and as a determinant of interest
rates through appropriate balance equations. These allow a central bank (respectively, a
money-market) to announce (respectively, to determine endogenously) interest rates in a
way that conserves the total money-supply and controls inflation.
General results are provided for the existence of such stationary equilibria, and several
explicitly solvable examples are treated in detail.
∗
Research supported by the Santa Fe Institute, and by the National Science Foundation under grants DMS–
97–32810 (Karatzas) and DMS–97–03285 (Sudderth). We are deeply indebted to the Referee and the Co-Editor
for the care with which they read the original version of the paper, and for their many helpful remarks.
1
Introduction
There are some relatively mundane aspects of economic activity which can easily be ignored
in an equilibrium theory concerned with the existence of prices but not with the mechanisms
which bring them into being. These are: (1) the presence of fiat money and the nature of the
conservation laws governing its supply in the markets and in the banking system; (2) the existence of the “float”, or a transactions need for money; (3) the need for default, bankruptcy and
reorganization rules, if lending is permitted; and (4) the nature of interest rates as parameters
or control variables or as endogenous variables.1 A process-model requires that these aspects be
explained and analyzed.
As in two previous papers [KSS1] (1994), and [KSS2] (1996), we study here an infinitehorizon strategic market game with a continuum of agents. The game models a simple economy
with one non-durable good (or “perishable commodity”) which is produced in the same quantity
in every period. The commodity endowments of individual agents are random, and fluctuate
from period to period. At the start of each period, agents must decide how much of their current
monetary wealth to spend on consumption of the commodity (“cash-in-advance” model). In
[KSS1] the only choice was between spending, and hoarding cash for the future. In [KSS2]
agents were able to borrow or lend money before spending, but were not allowed to borrow
more than they could pay back from their earnings in the next period. Since bankruptcy is a
prominent feature of real economies, we introduce here a more general model where bankruptcy
can and does occur.
The main focus of this paper is on a model with a Central Bank which makes loans and
accepts deposits. The bank sets two interest rates, one for borrowers and one for depositors.
Some unfortunate borrowers may not receive sufficient income to pay back their debts. To avoid
inflation, the bank must set the interest rate for borrowers sufficiently high, so that it will receive
enough money from high-income borrowers to offset the bad debts of the bankrupt, and also be
able to pay back depositors at a (possibly) different rate of interest. We assume that the bank
seeks not to make profit, but only to control inflation in the economy.
The rules of the game must, of course, specify the terms of bankruptcy. Almost every
conceivable rule seems to have occurred in history, but we have chosen for our model what
appears to be the simplest rule that can be analyzed mathematically. Namely, the bankrupt
receive a non-monetary “punishment” in units of utility, but are then forgiven their debts and
allowed to continue to play.
An interesting alternative to the model with a Central Bank is one with a money-market.
In this model, agents offer fiat money for lending, or bid I.O.U. notes for loans, and thereby
determine interest rates endogenously. Such a model is studied in [KSS2]. For the sake of clarity
and brevity, we shall concentrate here on the model with a central bank and limit ourselves to
a few remarks on the model with a money-market.
The game with a central bank, as we define it in Section 3, is a full-process model with
completely specified dynamics. Indeed, the game can be simulated for a finite number of players
1
A fifth key phenomenon is the velocity of money, which is defined as the volume of transactions per unit of
money per period of time. We delay our study of velocity and avoid the new difficulties it poses, by considering
discrete-time models where velocity is constrained to be between 0 and 1. The relationship between the discreteand continuous-time models is of importance. But we suspect that the transactions need for money cannot
be adequately modeled by continuous-time models alone, without superimposing some discrete-time aspects of
economic life on the continuous-time structure.
1
as was done for the model of [KSS1] by Miller and Shubik (1994). It would be very interesting
to obtain theorems about the limiting behavior of the model when it begins out of equilibrium,
and we do obtain some information (Lemma 7.5) about the extent to which the bank is able to
influence prices by its control of interest rates. However, we concentrate on the existence and
structure of a special type of equilibrium, called “stationary Markov equilibrium” (Theorems
5.1, 7.1, 7.2, and Examples 6.1-6.3), in which the price of the good and the distribution of wealth
among the agents remain constant.
In ongoing work, we propose to extend these results to a cyclical or stochastic overall
supply of goods per period, to examine the critical role played by a Central Bank in controlling
the money-supply, and to study the limitations in the Central Bank’s ability to control inflation
– as a function of the control variables it can utilize and of the frequency of its interventions.
We have limited our investigation to the trading of a single commodity. It appears that
the existence results may be extended to the case of several commodities; but in that context,
uniqueness will certainly not hold. Even with one commodity, if the system is started away from
the equilibrium distribution, we have been unable to establish general conditions for convergence
to equilibrium. We leave all these issues as open problems for further research.
2
Summary
In the next section, we provide a careful definition of the model under study, and also of the
notion of “stationary Markov equilibrium.” The key to our construction of such an equilibrium
is a detailed study in Section 4 of the one-person, dynamic programming problem faced by a
single player when the many-person model is in equilibrium (with constant price and interest
rates). We are then able to show in Section 5 that equilibrium occurs for given price, interest
rates, and wealth distribution, if two conditions hold: (i) the wealth distribution corresponds
to the aggregate of the invariant measures for the Markov chains associated with the wealth
processes of individual agents, and (ii) the bank balances its books by earning from borrowers
exactly what it owes its depositors. After a collection of illustrative examples in Section 6, fairly
general existence theorems are proved in Section 7 for the case of homogeneous agents. Section
8 offers a brief discussion of the model with a money-market (instead of a central bank).
3
The Model
Time in the economy is discrete and runs n = 1, 2, · · · . Uncertainty is captured by a probability
space (Ω, F, P) on which all the random variables of our model will be defined. There is a
continuum of agents α ∈ I , [0, 1], distributed according to a non-atomic probability measure ϕ
on the collection B(I) of Borel subsets of I. We regard such a non-atomic distribution of agents
as a fairly reasonable approximation of an anonymous mass-market, in which the actions of a
single agent do not have an appreciable effect on prices.
On each day, or time-period, n = 1, 2, ..., each agent α ∈ I receives a random endowment
Ynα (w) = Yn (α, w) in units of a single perishable commodity. The endowments Y1α , Y2α , ... for
a given agent α are assumed to be nonnegative, integrable, and independent, with common
distribution λα . We also assume that the variables Yn (α, w) are jointly measurable in (α, w), so
2
that the total endowment or “production”
Z
Qn (w) , Yn (α, w)ϕ(dα) > 0
is a well-defined, positive and finite random variable, for every n.
There is a loan-market and a commodity-market in each time-period t = n. For the loanmarket, the bank sets two interest rates, namely, r1n (w) = 1 + ρ1n (w) to be paid by borrowers
and r2n (w) = 1 + ρ2n (w) to be paid to depositors. In the commodity market, agents bid money
for consumption of the commodity, thereby determining its price pn (w) endogenously as will be
explained below. The interest rates are assumed to satisfy
1 ≤ r2n (w) ≤ r1n (w)
and
r2n (w) <
1
β
(3.1)
for all n ∈ N, w ∈ Ω, where β ∈ (0, 1) is a fixed discount factor.
Each agent α ∈ I has a utility function uα : R → R, which is assumed to be increasing and
concave with uα (0) = 0. For x < 0, uα (x) is negative and measures the “disutility” for agent
α of going bankrupt by an amount x; for x > 0, uα (x) is positive and measures the “utility”
derived from the consumption of x units of the commodity.
At the beginning of day t = n, the price of the commodity is pn−1 (w) (from the day
before) and the total amount of money held in the bank is Mn−1 (w). An agent α ∈ I enters the
α (w). If S α (w) < 0, then agent α has an unpaid debt from the previous
day with wealth Sn−1
n−1
α (w)/p
day, is assessed a non-monetary punishment of uα (Sn−1
n−1 (w)), is then forgiven the debt,
α
and plays the game from the wealth position 0. If Sn−1 (w) ≥ 0, then agent α has fiat money on
α (w). In both cases, an agent α (possibly after having been
hand and plays from position Sn−1
α (w))+ = max{S α (w), 0}.
punished and forgiven) will play from the wealth-position (Sn−1
n−1
α
Agent α also begins day n with information Fn−1
⊂ F, a σ-algebra of events that measures
past prices pk , past total endowments Qk and interest-rates r1k , r2k , as well as past individual
wealth-levels, endowments, and actions S0α , Skα , Ykα , bαk , for k = 1, ..., n − 1. (It may, or may
not, measure the corresponding quantities for other agents.) Based on this information, agent
α bids an amount
α
bαn (w) ∈ [0, (Sn−1
(w))+ + k α ]
(3.2)
of fiat money for the commodity on day n. The constant k α ≥ 0 is an upper bound on allowable
loans. It is assumed that the mapping (α, w) 7→ bαn (w) is B(I) ⊗ Fn−1 −measurable, where
_
α
Fn−1 ,
Fn−1
α
α
is the smallest σ-algebra containing Fn−1
for all α ∈ I. Consequently, the total bid
Z
Bn (w) , bαn (w)ϕ(dα) > 0
(3.3)
is a well-defined random variable, assumed to be strictly positive.
After the price pn (w) for day t = n has been formed, according to the rule of (3.12) below,
each agent α receives his bid’s worth xαn (w) , bαn (w)/pn (w) of the commodity, consumes it in
3
the same period (the “perishable” nature of the commodity), and thereby receives uα (xαn (w)) in
utility. The total utility that agent α receives during the period is thus
¾
½ α α
α (w) ≥ 0
if Sn−1
u (xn (w)),
α
ξn (w) ,
.
(3.4)
α (w)/p
α
uα (xαn (w)) + uα (Sn−1
n−1 (w)), if Sn−1 (w) < 0
The
payoff for agent α during the entire duration of the game is the discounted sum
P∞ total
n−1
ξnα (w).
n=1 β
3.1
Strategies
A strategy π α for an agent α specifies the bids bαn as random variables that satisfy (3.2) and are
α −measurable (thus also F
Fn−1
n−1 −measurable), for every n ∈ N. A collection Π = {πα , α ∈ I}
of strategies for all the agents is admissible if, for every n ∈ N, the mapping (α, w) 7→ bαn (w)
is Gn−1 ≡ B(I) ⊗ Fn−1 −measurable. We shall always assume that the collection of strategies
played by the agents is admissible. Consequently, the macro-variable Bn (w) of (3.3), which
represents the total bid in period n, is well-defined and Fn−1 −measurable.
3.2
Dynamics
In order to explain the dynamics of the model, we concentrate again on day t = n. After the bids
for this day have been made and the price pn (w) has been formed, according to the rule of (3.12)
below, the agents’ endowments Ynα (w) are revealed and each agent α receives his endowment’s
worth pn (w)Ynα (w) in fiat money, according to the day’s price. Now there are three possible
situations for agent α on day n:
(i) Agent α is a depositor: this means that α’s bid bαn (w) is strictly less than his wealth
α (w))+ = S α (w) and he deposits (or lends) the difference
(Sn−1
n−1
¡ α
¢+
α
`αn (w) , Sn−1
(w) − bαn (w) = Sn−1
(w) − bαn (w).
(3.5)
α (w))+ .) At the end of the day, α gets back his
(We set `αn (w) equal to 0 if bαn (w) ≥ (Sn−1
deposit with interest, as well as his endowment’s worth in fiat money, and thus moves to
the new wealth level
Snα (w) , r2n (w)`αn (w) + pn (w)Ynα (w) > 0.
(3.6)
α (w))+ , so
(ii) Agent α is a borrower: this means that α’s bid bαn (w) exceeds his wealth (Sn−1
he must borrow the difference
α
dαn (w) , bαn (w) − (Sn−1
(w))+ .
(3.7)
α (w))+ .) At the end of the day, α owes the bank
(We set dαn (w) equal to 0 if bαn (w) ≤ (Sn−1
α
r1n (w)dn (w), and his new wealth position is
Snα (w) , pn (w)Ynα (w) − r1n (w)dαn (w),
a quantity which may be negative. Agent α is then required to pay back, from his endowment pn (w)Ynα (w), as much of his debt r1n (w)dαn (w) as he can. Thus, agent α pays back
the amount
hαn (w) , min{r1n (w)dαn (w), pn (w)Ynα (w)}
(3.8)
and his cash holdings at the end of the period are (Snα (w))+ = pn (w)Ynα (w) − hαn (w).
4
(iii) Agent α neither borrows nor lends: in this case the agent bids his entire cash-holdings
α (w))+ and ends the day with exactly his endowment’s worth in fiat money
bαn (w) = (Sn−1
Snα (w) = pn (w)Ynα (w) ≥ 0.
Using the notation of (3.5)–(3.8) we can write a single formula for α’s wealth position at the
end of the period
Snα (w) = pn (w)Ynα (w) + r2n (w)`αn (w) − r1n (w)dαn (w),
(3.9)
and another formula for α’s cash-holdings
(Snα (w))+ = pn (w)Ynα (w) + r2n (w)`αn (w) − hαn (w).
(3.10)
The wealth position Snα (w) may be negative, but the amount (Snα (w))+ of holdings in cash is,
of course, nonnegative.
3.3
The Conservation of Money
Let Mn (w) be the total quantity of fiat money held by the bank, and
Z
M̃n (w) , (Snα (w))+ ϕ(dα)
(3.11)
be the total amount of fiat money held by the agents, at the end of the time-period t = n. Thus,
the total wealth in fiat money in the economy is Wn (w) , Mn (w) + M̃n (w). Consider the simple
rule
Bn (w)
,
(3.12)
pn (w) =
Qn (w)
which forms the commodity price as the ratio of the total bid to total production. This rule
turns out to be both necessary and sufficient for the conservation of money. For the rest of the
paper we assume that (3.12) holds, and thus money is conserved.2
Lemma 3.1 The total quantity Wn (w) = Mn (w) + M̃n (w) of fiat money in the economy is the
same for all n and w, if and only if (3.12) holds.
¡ α
¢+ ¡
¢
α (w))+ + =
Proof: Use (3.5)–(3.11) to see that `αn (w)−dαn (w) = (Sn−1
(w))+ − bαn (w) − bαn (w) − (Sn−1
α (w))+ − bα (w), and check that (3.12) implies then
(Sn−1
n
Z
Mn (w) − Mn−1 (w) = [`αn (w) − dαn (w) + hαn (w) − r2n (w)`αn (w)]ϕ(dα)
Z
Z
α
+
α
= [(Sn−1 (w)) − bn (w)]ϕ(dα) + [hαn (w) − r2n (w)`αn (w)]ϕ(dα)
Z
Z
α
= M̃n−1 (w) − pn (w) Yn (w)ϕ(dα) − [r2n (w)`αn (w) − hαn (w)]ϕ(dα)
= M̃n−1 (w) − M̃n (w).
2
The combination of the assumption of a fixed time-grid for payments, together with a conservation law for
fiat money, apparently give us a quantity theory of money, as the velocity of money is constrained to be less than
or equal to one. Superficially, we have: Price × Quantity = Money × Velocity (P Q = M V ), but this artifact of
our model is for simplicity. Velocity can, and is, varied both by strategic delays in payments and by the creation
and use of different financial instruments which are close money-substitutes.
5
Reasoning in the opposite direction, we see that (3.12) is also a necessary condition for the
conservation principle Mn (w) + M̃n (w) = Mn−1 (w) + M̃n−1 (w) to hold.
¥
3.4
Equilibrium with Exogenous Interest Rates
Interest rates are announced by the Central Bank, and can be viewed as exogenous in our model.
(An interesting question for future research is how and to what extent the bank can control prices
by its choice of interest rates. Lemma 7.5 below can be viewed as a first step in this direction.)
In equilibrium agents must be optimizing given a rational forecast of interest rates and prices.
Let {r1n , r2n , pn }∞
n=1 be a given system of interest rates and prices. The total expected
utility to an agent α from a strategy π α when S0α = s, is given by
I α (π α )(s) , E
∞
X
β n−1 ξnα (w),
n=1
and his optimal reward is
V α (s) , sup I(π α )(s).
πα
Definition 3.1 An equilibrium is a system of interest rates and prices {r1n , r2n , pn }∞
n=1 and an
admissible collection of strategies {π α , α ∈ I} such that
(i) the prices {pn }∞
n=1 satisfy (3.12), and
(ii) For every α ∈ I, we have I α (π α )(S0α ) = V α (S0α ) when every other agent β ∈ I\{α}
plays according to the strategy π β .
Observe that we place no restrictions on interest rates in this definition. We are assuming
implicitly that the bank sets interest rates arbitrarily, and has enough cash to cover all demands
for loans and to meet all depositor requirements, in each period.
In this paper, we will not study the existence and structure of an equilibrium as general
as that of Definition 3.1. We shall concentrate instead on the special case of a stationary
equilibrium, which will be defined momentarily.
3.5
The Distribution of Wealth
An admissible collection of strategies {π α , α ∈ I} together with an initial distribution for {S0α ,
α ∈ I} determines the random measures
Z
νn (A, w) , 1A (Snα (w))ϕ(dα), A ∈ B(R)
(3.13)
that describe the distribution of wealth across agents for n = 0, 1, . . . .
3.6
Stationary Equilibrium
In order to obtain a stationary equilibrium, we must have a stationary economy. Thus, we shall
assume from now on that the total production Qn (w) is constant, namely
Z
Q = Yn (α, w)ϕ(dα) > 0, for every w ∈ Ω, n ∈ N.
(3.14)
6
A simple technique of Feldman and Gilles (1985) allows us to construct B(I) ⊗ F −measurable
functions
(a, w) 7→ Ynα (w) = Yn (α, w) : I × Ω → [0, ∞)
(3.15)
for n ∈ N, which have the desired properties.
Remark 3.1 If, in particular, all the distributions λα ≡ λ (∀α ∈ I) are the same, Feldman and
Gilles (1985) show that the sequence of measurable functions (3.15) can be constructed in such
a way that
(a) for every given α ∈ I, the random variables Y1α (·), Y2α (·), ... are independent with common
distribution λ,
(b) for every given w ∈ Ω, the measurable functions Y1• (w), Y2• (w), ... are independent with
common distribution λ, and
(c) (3.14) holds.
Thus, in this case, Q =
R
yλ(dy) > 0.
¥
α
Definition 3.2 A stationary Markov equilibrium is an equilibrium {r1n , r2n , pn }∞
n=1 , {π , α ∈
I} such that, in addition to conditions (i) and (ii) of Definition 3.1, the following are satisfied:
(iii) the interest rates r1n (w), r2n (w) and prices pn (w) have constant values r1 , r2 , and p,
respectively;
(iv) the wealth distributions νn (·, w) are equal to a constant measure µ;
(v) the quantities Mn (w) and M̃n (w), corresponding to money held by the bank and by the
agents, have constant values M and M̃ , respectively; and
(vi) each agent α ∈ I follows a stationary Markov strategy π α , which means that the bids bαn
specified by π α can be written in the form
α
bαn (w) = cα ((Sn−1
(w))+ ), for every w ∈ Ω, n ∈ N.
Here cα : [0, ∞) → [0, ∞) is a measurable function such that 0 ≤ cα (s) ≤ s + k α for every
s ≥ 0.
The conditions (v) in Definition 3.2 are redundant, as is made clear by the following lemma.
Lemma 3.2 In any equilibrium, conditions (i) and (iv) imply (v), and conditions (iv) and (v)
imply (i).
R
R
R
Proof: If (iv) holds, then M̃n (w) = (Snα (w))+ ϕ(dα) = s+ νn (ds, w) = s+ µ(ds) is the same
for all n and w. Thus both assertions follow from Lemma 3.1.
¥
If our model is in stationary Markov equilibrium, then an individual agent faces a sequential
optimization problem with fixed price and fixed interest rates. After a detailed study of this
one-person game in the next section, we shall return to the many-person model in Section 5.
7
4
The One-Person Game
Suppose that the model of the previous section is in stationary Markov equilibrium, and let us
focus on the optimization problem facing a single agent. (We omit the superscript α in this
section.) As we will now explain, this problem is a discounted dynamic programming problem
in the sense of Blackwell (1965).
The interest rates r1 = 1 + ρ1 , r2 = 1 + ρ2 , and the discount factor β are assumed to
satisfy (3.1) as before:
1 ≤ r2 ≤ r1 and r2 < 1/β.
(4.1)
The state space S represents the possible wealth-positions for the agent. Because the
nonnegative number k is an upper bound on loans, the agent never owes more than r1 k. Thus
we can take S = [−r1 k, ∞). The price p ∈ (0, ∞) remains fixed throughout. The agent’s utility
function u : R → R is, as before, concave and increasing with u(0) = 0.
In each period the agent begins at some state s ∈ S. If s < 0, the agent is punished by the
amount u(s/p) and is then allowed to play from state s+ = 0. If s ≥ 0, the agent chooses any
action or bid b ∈ [0, s + k], purchases b/p units of the commodity, and receives u(b/p) in utility.
In the terminology of dynamic programming, the action set is B(s+ ), where B(s) = [0, s+k] for
s ≥ 0, and the daily reward of an agent at state s who takes action b ∈ B(s) is
½
¾
u(b/p),
s≥0
r(s, b) =
.
u(s/p) + u(b/p), s < 0
The remaining ingredient is the law of motion that specifies the conditional distribution q(·|s, b)
of the next state s1 by the rule
½
¾
−r1 (b − s+ ) + pY, s+ ≤ b
s1 =
.
r2 (s+ − b) + pY,
s+ ≥ b
Here Y is a nonnegative, integrable random variable with distribution λ. For ease of notation,
we introduce the concave function
½
¾
r1 x, x ≤ 0
g(x) ,
≡ g(x; r1 , r2 ).
(4.2)
r2 x, x ≥ 0
Then the law of motion becomes s1 = g(s+ − b) + pY.
A player begins the first day at some state s0 and selects a plan π = (π1 , π2 , ...), where πn
makes a measurable choice of the action bn ∈ B(sn−1 ) as a function of (s0 , b1 , s1 , ..., bn−1 , sn−1 ).
A plan π, together with the law of motion, determine the distribution of the stochastic process
s0 , b1 , s1 , b2 , ... of states and actions. The return from π is the function
I(π)(s) , Eπs0 =s
∞
X
β n r(sn , bn+1 ), s ∈ S.
(4.3)
n=0
The optimal return or value function is
V (s) , sup I(π)(s),
π
s ∈ S.
A plan π is called optimal for the one-person game, if I(π) = V .
8
(4.4)
If the utility function u(·) is bounded, then so is r(·, ·), and our player’s optimization
problem is a discounted dynamic programming problem as in Blackwell (1965). In the general
case, because u(·) is concave and increasing, we have
u(−r1 k) ≤ u(s) ≤ u(s+ ) ≤ u0+ (0)s+
(4.5)
for all s ∈ S. This domination by a linear function is sufficient, as it was in [KSS1] and [KSS2],
for many of Blackwell’s results to hold in our setting as well. Thus V (·) satisfies the Bellman
equation
£
¤
V (s) = sup r(s, b) + β · EV (g(s+ − b) + pY )
b∈B(s)
(
=
)
sup {u(b/p) + β · EV (g(s − b) + pY )}; s ≥ 0
0≤b≤s+k
u(s/p) + V (0);
s<0
Equivalently, V = T V where T is the operator
£
¤
(T ψ)(s) , sup r(s, b) + β · Eψ(g(s+ − b) + pY ) ,
.
(4.6)
(4.7)
b∈B(s+ )
defined for measurable functions ψ : S → R that are bounded from below.
A plan π is called stationary if it has the form bn = c(s+
n−1 ) for all n ≥ 1, where c :
[0, ∞) → [0, ∞) is a measurable function such that c(s) ∈ B(s) for all s ≥ 0. We call c(·) the
consumption function for the stationary plan π.
The following characterization of optimal stationary plans, given by Blackwell (1965),
extends easily to our situation, so we omit the proof.
Theorem 4.1 For a stationary plan π with consumption function c(·), the following conditions
are equivalent:
(a) I(π) = V .
(b) V (s) = r(s, c(s+ )) + β · EV (g(s+ − c(s+ )) + pY ),
s ∈ S.
(c) T (I(π)) = I(π).
Under our assumptions, a stationary optimal plan exists but need not be unique. However,
if the utility function u(·) is smooth and strictly concave, there is a unique optimal plan and the
next theorem has some information about its structure.
For the rest of this section, we make the following assumption:
Assumption 4.1 The utility function u(·) is concave and strictly increasing on S, strictly concave on [0, ∞), differentiable at all s 6= 0, and we have u(0) = 0, u0+ (0) > 0.
Theorem 4.2 Under Assumption 4.1, the following hold:
(a) The value function V (·) is concave, increasing.
(b) There is a unique optimal stationary plan π corresponding to a continuous consumption
function c : [0, ∞) → (0, ∞) such that c(s) ∈ B(s) for all s ∈ [0, ∞). Furthermore, the
functions s 7→ c(s) and s 7→ s − c(s) are nondecreasing.
9
(c) For s ∈ S \ {0}, the derivative V 0 (s) exists and
(
)
1 0
0
p u (c(s)/p); s > 0
V (s) =
.
1 0
p u (s/p); s < 0
(d) For s > 0, we have c(s) > 0. If βr1 < 1, then c(0) > 0.
(e) lims→∞ c(s) = ∞.
(f) If the limit L , lims→∞ (s − c(s)) is positive, it satisfies the equation
£
¤
u0 (∞)
= E u0 (c(r2 L + pY ) / p)
βr2
where u0 (∞) , limx→∞ u0 (x) = inf x∈R u0 (x); in particular, L < ∞ if u0 (∞) > 0. If,
furthermore, c(·) is strictly increasing and u0 (∞) > 0, then L is uniquely determined in
(0, ∞) by this equation.
(g) If L > 0 as in (f ), then s∗ , inf{s > 0 : s > c(s)} is characterized by the equation
u0 (s∗ /p) = βpr2 · E[V 0 (pY )]
and we have s∗ > pε0 , where ε0 , sup{ε > 0 : P[Y ≥ ε] = 1}.
Part (b) of the theorem asserts, inter alia, that under optimal play, an agent both consumes
more and deposits more money, as his wealth increases. Part (c) is a version of the “envelope
equation.” Part (d) says that an agent with a positive amount of cash always spends a positive
amount. However, because we have imposed no upper bound on the interest rate r1 , it could
happen that c(s) ≤ s for all s, or, equivalently, that no borrowing occurs. Part (d) further
asserts that if βr1 < 1, then there will be an active market for borrowing money.
The proof of Theorem 4.2 is presented in the following subsection. It is a bit lengthy;
impatient readers may prefer to skip or skim it.
4.1
The Proof of Theorem 4.2
The n-day value function Vn (·) represents the best a player can do in n days of playing the
one-person game of (4.3)-(4.4). It can be calculated by the induction algorithm:
½
¾
u((s + k)/p),
s≥0
V1 (s) = (T 0)(s) =
,
u(s/p) + u(k/p), s < 0
Vn+1 (s) = (T Vn )(s),
s ∈ S, n ≥ 1.
(4.8)
Furthermore, it is not difficult to show, with the aid of (4.5), that
lim Vn (s) = V (s), s ∈ S.
n→∞
(4.9)
The idea of the proof of Theorem 4.2 is to derive properties of the Vn (·) by a recursive
argument based on (4.8), and then to deduce the desired properties of V (·). For the recursive
argument, we consider functions w : S → R satisfying the following condition.
10
Condition 4.1 The function w(·) is concave, increasing on S, differentiable on S \ {0}, with
0 (0) ≤ 1 u0 (0), and w(s) = u(s/p) + w(0) for s ≤ 0.
w+
p +
Proposition 4.1 If w(·) satisfies Condition 4.1, then so does T w(·).
Corollary 4.1 For n ≥ 1, Vn (·) satisfies Condition 4.1.
Proof: Observe from (4.8) that V1 (·) satisfies the condition, and then apply the proposition
and (4.8) again.
¥
For the proof of Proposition 4.1 (established through a series of Lemmata 4.2-4.5 below), fix a
concave, increasing function w(·) and set
ψs (b) = ψ(s, b) , u(b/p) + β · Ew(g(s − b) + pY ); s ≥ 0, 0 ≤ b ≤ s + k.
(4.10)
Lemma 4.1 Suppose that w : S → R is concave, increasing.
(a) For each s ≥ 0, ψs (·) is strictly concave on [0, s + k] and attains its maximum at a unique
point c(s) ≡ cw (s).
(b) (s, b) 7→ ψ(s, b) is a concave function on the convex, two-dimensional set {(s, b) : s ≥ 0,
0 ≤ b ≤ s + k}.
Proof: Elementary, using the facts that g(·) of (4.2) and w(·) are concave, and u(·) is strictly
concave on [0, ∞); recall Assumption 4.1.
¥
Now define v(s) ≡ vw (s) , (T w)(s) for s ∈ S. By Lemma 4.1(a), we can write
½
¾
u(cw (s)/p) + β · Ew(g(s − cw (s)) + pY ), s ≥ 0
vw (s) =
.
u(s/p) + vw (0),
s<0
(4.11)
It may be helpful to think of vw (·) as the optimal return when an agent plays the game for one
day and receives a terminal reward of w(·).
Lemma 4.2 Suppose that w : S → R satisfies Condition 4.1. Then the function cw (·) has the
following properties:
(a) Both s 7→ cw (s) and s 7→ s − cw (s) are nondecreasing.
(b) cw (·) is continuous.
(c) For s > 0, we have cw (s) > 0.
(d) If βr1 < 1, then cw (0) > 0.
(e) lims→∞ cw (s) = ∞.
11
Proof: For (a), note that the function w̃(·) , β Ew(g(·) + pY ) is concave, and thus the problem
of maximizing ψs (b) = u(b/p) + w̃(s − b) over b ∈ [0, s + k] is a standard allocation problem
for which (a) is well-known (see, for example, Theorem I.6.2 of Ross (1983)). Property (b)
follows from (a). For (c), let s > 0; use Condition 4.1, the definition of ψs (·) in (4.10), and
0 (r s + pY ) ≥
our standing assumption βr2 < 1, to see that p · (ψs )0+ (0) = u0+ (0) − βr2 p · Ew−
2
0
0
u+ (0) − βr2 p · w+ (0) > 0. To prove (d) notice that, for s = 0, the same calculation works with
r2 replaced by r1 . This is because of the definition of g(·) in (4.2). For the final assertion, see
the proof of Theorem 4.3 in [KSS1], which relies crucially on the strict increase of u(·).
¥
The proof that vw (·) is concave will take three steps. The first is to show that vw (·) is
concave except possibly at the origin.
Lemma 4.3 The function vw = T w is concave on [−r1 k, 0] and also on [0, ∞).
Proof: The concavity of vw (·) on [−r1 k, 0] is clear from (4.11) and the concavity of u(·). For
s ≥ 0, vw (s) = sup{ψ(s, b) : 0 ≤ b ≤ s + k} is the supremum of a concave function (cf. Lemma
4.1(b)) over a convex set. It is well-known that such an operation yields a concave function. ¥
The second step is a version of the “envelope equation.”
Lemma 4.4 For s 6= 0, the function vw (·) is differentiable at s and
(
)
1 0
0
p u (cw (s)/p), s > 0
vw (s) =
.
1 0
s<0
p u (s/p),
Proof: For s < 0, the assertion is obvious from (4.11). Let us then fix s > 0 and, for simplicity,
write
³ v(·)´ for v³w (·) ´and c(·) for cw (·). Note first that, for ε > 0, we have v(s + ε) − v(s) ≥
− u c(s)
, since an agent with wealth s + ε can spend c(s) + ε and then be in
u c(s)+ε
p
p
0 (s) ≥
the same position as an agent with s who spends the optimal amount c(s). Hence, v+
1 0
other hand, recall (c) and observe, for 0 < ε < s ∧ c(s), that we have
p u (c(s)/p). On the
³
´
³
´
c(s)
v(s) − v(s − ε) ≤ u p − u c(s)−ε
, since an agent with wealth s − ε can spend c(s) − ε and
p
0 (s) ≤ 1 u0 (c(s)/p).
then be in the same position as an optimizing agent starting at s. Hence, v−
p
0
0
Finally, v− (s) ≥ v+ (s) because, by Lemma 4.3, v(·) is concave on [0, ∞).
¥
Lemma 4.5 The function vw (·) is concave on S, and (vw )0+ (0) ≤ p1 u0+ (0).
Proof: By Lemma 4.3, it suffices for concavity to show that (vw )0+ (0) ≤ (vw )0− (0). But by
Lemma 4.4 and (4.11), we have (vw )0+ (0) = lims↓0 p1 u0 (cw (s)/p) = p1 u0+ (cw (0)/p) ≤ p1 u0+ (0) ≤
1 0
0
¥
p u− (0) = (vw )− (0).
Proposition 4.1 follows from (4.11) and Lemmata 4.4 and 4.5. We are finally prepared to
complete the proof of Theorem 4.2.
Proof of Theorem 4.2: From Corollary 4.1, the n-day value functions Vn (·) are concave,
increasing; by (4.9), they converge pointwise to V (·). Hence, V (·) is also concave and increasing.
12
By Lemma 4.1(a) with w(·) = V (·) and (4.6), there is for each s ≥ 0 a unique c(s) ∈ [0, s + k]
such that V (s) = u(c(s)/p)+β ·EV (g(s−c(s))+pY ). Set formally c(s) ≡ c(0) for −r1 k ≤ s < 0,
and it follows from Theorem 4.1 that c(·) corresponds to the unique optimal stationary plan.
Next, let cn (·) ≡ cVn (·), n ≥ 1, where cVn (·) is the notation introduced in Lemma 4.1(a).
It can be shown using the techniques of Schäl (1975), or by a direct argument, that cn (s) → c(s)
as n → ∞ for each s ∈ S. Thus, the functions c(·) and s 7→ s − c(s) are nondecreasing, because
the same is true of cn (·) and s 7→ s − cn (s) for each n. In particular, c(·) in continuous. Now by
Lemma 4.4, we can write
Z s
p · [Vn+1 (s) − Vn+1 (0)] =
u0 (cn (x)/p)dx, s ≥ 0, n ∈ N,
0
Rs
and let n → ∞, to obtain p · [V (s) − V (0)] = 0 u0 (c(x)/p)dx, s ≥ 0. Differentiate to get part
(c) of the theorem for s > 0. For s < 0, use (4.6).
Let s ↓ 0 in (c) to get pV+0 (0) = u0+ (c(0)/p) ≤ u0+ (0). Thus the value function V satisfies
Condition 4.1. By the Bellman equation (4.6), V = T V = vV in the notation of (4.11) with
c = cV . Thus, parts (d) and (e) of the theorem follow from Lemma 4.2.
For parts (f) and (g), note that for all s > 0 such that s > c(s), we have
£
¤
u0 (c(s)/p) = βpr2 · E V 0 (r2 (s − c(s)) + pY ) .
This leads to the characterization for s∗ in part (g), as well as to
1 0
u (c(s)/p) ≤ p · E[V 0 (pY )] = E[u0 (c(pY )/p)],
βr2
for s > 0, s > c(s)
thanks to (c). Therefore, letting s → ∞ we obtain, in conjunction with (b), (c) and (e):
u0 (∞) ≤ βr2 · E[u0 (c(pY )/p)] and
£
¤
u0 (∞)
= p · E[V 0 (Lr2 + pY )] = E u0 (c(r2 L + pY )/p) .
βr2
Suppose now that u0 (∞) > 0, and that c(·) is strictly increasing; then the function f (·) ,
E [u0 (c(· r2 + pY )/p)] is strictly decreasing with f (0+) = E[u0 (c(pY )/p)] ≥ u0 (∞)/βr2 and
f (∞) = u0 (∞); thus, there is a unique root ` ∈ (0, ∞) of the equation f (`) = u0 (∞)/βr2 . Finally, for the inequality of (g), notice that we have
1
1
V 0 (s∗ ) = u0 (c(s∗ )/p) = u0 (s∗ /p) = βr2 · E[V 0 (pY )] < E[V 0 (pY )] ≤ V 0 (pε0 ),
p
p
and the inequality s∗ > pε0 follows from the decrease of V 0 (·) on (0, ∞).
4.2
¥
The Wealth-Process of an Agent
Suppose now that an agent begins the one-person game of this section with wealth S0 = s0 and
follows the stationary plan π of Theorem 4.2. The process {Sn }n∈N of the agent’s successive
wealth-levels then satisfies the rule
+
+
Sn = g(Sn−1
− c(Sn−1
)) + pYn ,
13
n ≥ 1,
(4.12)
where the endowment variables Y1 , Y2 , ... are independent with common distribution λ, and g(·)
is the function of (4.2). Hence, {Sn }n∈N is a Markov chain with state-space S = [−kr1 , ∞). An
understanding of this Markov chain is essential to an understanding of the many-person game
of Section 3. In particular, it is important to know when the chain has an invariant distribution
with finite mean.
Theorem 4.3 Under Assumption 4.1, the Markov chain {Sn }n∈N of (4.12) has an invariant
distribution with finite mean, if either one of the following conditions holds:
(a) u0 (∞) > 0,
(b) r2 = 1
and
R
y 2 λ(dy) < ∞.
Under the conditions (b), this invariant distribution is unique.
Sketch of Proof: (a) If u0 (∞) > 0 one can show, as in Theorem 4.2(f) or as in Corollary
3.6 of [KSS2], that the function g(s+ − c(s+ )), s ∈ S is bounded, and then complete the
proof as in Proposition 3.7 of [KSS2]. For (b), one applies results of Tweedie (1988) as in
the proof of Proposition 3.8 of [KSS2]. For the uniqueness result under condition (b), one
applies the arguments in the proof of Theorem 5.1 in [KSS2], pp. 992-994 with only very minor
modifications, and taking advantage of the fact that the interval R = [−r1 k, s∗ ] is a regeneration
set for the Markov chain of (4.12). As in that proof, one shows that this set can be reached in
finite expected time, using Theorem 4.2(e,g).
¥
Remark 4.1 For a given vector θ = (r1 , r2 , p) of interest rates as in (4.1) and price p ∈ (0, ∞),
we denote by cθ (·) ≡ c(·; θ), µθ (·) ≡ µ(·; θ) the optimal consumption function of Theorem 4.2
and the invariant measure of Theorem 4.3, respectively. If the bound on loans k is a function of
θ such that k(θ) = k(r1 , r2 , p) = p k(r1 , r2 , 1), then, as in (4.4) and (4.6) of [KSS1], we have
the scaling properties
¶
µ
s
; r1 , r2 , 1
(4.13)
cθ (s) ≡ c(s; r1 , r2 , p) = p · c
p
µ
¶
ds
µθ (ds) ≡ µ(ds; r1 , r2 , p) = µ
; r1 , r2 , 1 .
(4.14)
p
5
Conditions for Stationary Markov Equilibrium
We shall return in this section to the Strategic Market Game of Section 3 and show how to
construct a stationary Markov equilibrium (Definition 3.2) for this game, using the basic building
blocks of Section 4. This construction will rest on two basic assumptions (cf. Assumptions 5.1
and 5.2 below):
(i) Each agent uses a stationary plan which is optimal for his own (one-person) game, and
for which the associated Markov chain of wealth-levels (4.12) has an invariant distribution with
finite mean.
(ii) The bank “balances its books”, that is, selects r1 , and r2 in such a way that it pays
back (in the form of interest to depositors, and of loans to borrowers) what it receives (in the
form of repayments with interest, from borrowers).
14
The construction is significantly simpler, at least analytically if not conceptually, when
all the agents are “homogeneous”, i.e., when they all have the same utility function uα ≡ u,
income distribution λα ≡ λ, and upper bound on loans k α ≡ k, ∀α ∈ I. We shall deal with this
case throughout, but refer the reader to [KSS1] and [KSS2] for aggregation techniques that can
handle countably many types of homogeneous agents (and can be used in our present context
as well).
Let us fix a price p ∈ (0, ∞) for the commodity, and two interest rates r1 = 1 + ρ2 (from
borrowers) and r2 = 1 + ρ2 (to depositors) as in (4.1).
Assumption 5.1 The one-person game of Section 4 has a unique optimal plan π corresponding
to a continuous consumption function c : [0, ∞) → [0, ∞), and the associated Markov chain of
wealth-levels in (4.12) has an invariant distribution µ on B(S) with
Z
sµ(ds) < ∞.
(5.1)
Assumption 5.2 Under this invariant distribution µ of wealth-levels, the bank balances its
books, in the sense that the total amount paid back by borrowers equals the sum of the total
amount they borrowed, plus the total amount that the bank pays to lenders in the form of interest:
ZZ
Z
Z
+
+
{py ∧ r1 d(s )}µ(ds)λ(dy) = d(s )µ(ds) + ρ2 `(s+ )µ(ds).
(5.2)
Here we have denoted by
d(s) , (c(s) − s)+ ,
`(s) , (s − c(s))+
(5.3)
the amounts of money borrowed and deposited, respectively, under optimal play in the oneperson game, by an agent with wealth-level s ≥ 0.
Theorems 4.2 and 4.3 provide sufficient conditions for Assumption 5.1 to hold. We shall
derive in Section 7 similar, though somewhat less satisfactory, sufficient conditions for Assumption 5.2. In Section 6 we shall present several examples that can be solved explicitly. If the
initial wealth distribution ν0 is equal to µ, and if every agent uses the plan π, then equation
(5.2) just says that the quantities M0 , M1 , ... of money held by the bank in successive periods
are equal to a constant as in Definition 3.2(v). Thus, Assumption 5.2 is a necessary condition
for the existence of a stationary Markov equilibrium.
Lemma 5.1 Under the Assumptions 5.1 and 5.2, we have
Z
1
c(s+ )µ(ds).
p=
Q
(5.4)
Proof. From (4.12) and (5.2) we obtain S1 = g(S0+ −c(S0+ ))+pY1 = pY1 +r2 `(S0+ )−r1 d(S0+ ), so
that
E(S1+ ) = E[(pY1 + r2 `(S0+ ))1{d(S + )=0} ] + E[(pY1 − r1 d(S0+ ))1{0<r1 d(S + )≤pY1 } ]
0
0
= p · E(Y1 ) − E[pY1 · 1{r1 d(S + )>pY1 } ] +
0
= pQ +
r2 E`(S0+ )
− E[pY1 ∧
r1 d(S0+ )].
15
r2 E[`(S0+ )]
−
E[r1 d(S0+ )1{r1 d(S + )≤pY1 } ]
0
Here S0 and Y1 are independent random variables with distributions µ and λ, respectively. From
(5.2), the last expectation above is E[pY1 ∧ r1 d(S0+ )] = E[d(S0+ ) + ρ2 `(S0+ )], so that
E(S1+ ) = pQ + E[`(S0+ ) − d(S0+ )] = pQ − Ec(S0+ ) + E(S0+ ).
But, from Assumption 5.1, S1 has the same distribution as S0 (namely µ), so that in particular
E(S0+ ) = E(S1+ ), and thus p = E[c(S0+ )]/Q.
¥
Theorem 5.1 Suppose that for fixed interest rates r1 , r2 as in (4.1), we can find a price
p ∈ (0, ∞) such that the consumption function c(·) ≡ cθ (·) and the probability measure µ ≡
µθ (notation of Remark 4.1 with θ = (r1 , r2 , p)) satisfy the Assumptions 5.1 and 5.2. Let π
be the corresponding optimal stationary plan; then the family Π = {π α }α∈I , πα = π (∀α ∈ I)
results in a stationary Markov equilibrium (p, µθ ) with θ = (r1 , r2 , p).
Remark 5.1 From the scaling properties (4.13), (4.14) and from (5.2), it is clear that if the
procedure of Theorem 5.1 leads to Stationary Markov equilibrium for some p ∈ (0, ∞), then it
does so for every p ∈ (0, ∞). For a given, constant level W0 of total wealth in the economy, we
can then determine the “right” price p# ∈ (0, ∞) via
Z
Z
α
+
W0 − M0 = (S0 (w)) ϕ(dα) = s+ ν0 (ds, w)
¶
µ
Z
Z
ds
+
+
; r1 , r2 , 1 ,
=
s µ(ds; r1 , r2 , p# ) = s µ
p#
namely as
Z
p# = (W0 − M0 ) /
s+ µ(ds; r1 , r2 , 1).
(5.5)
Recall (3.12) and the discussion following it, as well as (4.14).
Proof of Theorem 5.1. From Remark 3.1, the Markov Chain
α
α
Snα (w) = g((Sn−1
(w))+ − cθ ((Sn−1
(w))+ ) + pYnα (w), n ∈ N
of (4.12) has the same dynamics for each fixed α ∈ I, as for each fixed w ∈ Ω. In particular,
µ = µθ is a stationary distribution for the chain {Snα (·)}n∈N for each given α ∈ I, as well as for
the chain {Sn• (w)}n∈N for each given w ∈ Ω.
Assume that the initial price is p0 = p ∈ (0, ∞), and that the initial wealth-distribution
ν0 of (3.13) with n = 0, is ν0 = µ ≡ µθ . Then from (3.12) and (5.4),
Z
Z
1
1
p1 (w) =
cθ ((Snα (w))+ )ϕ(dα) =
c(s+ )µ(ds) = p.
Q I
Q S
On the other hand, since µ is invariant for the chain, we have ν1 = µ as well. By induction,
pn = p and νn = µ (∀n ∈ N).
Condition (i) of Definition 3.1 is true by assumption, and we have verified (iii) and (iv)
of Definition 3.2, whereas (vi) holds by our choice of π α = π. Condition (ii) of Definition 3.1
follows from the optimality of π in the one-person game and from the fact that a change of
strategy by a single player cannot alter the price. Condition (v) of Definition 3.2 follows from
Lemma 3.2.
¥
16
6
Examples
We consider in this section three examples, for which the one-person game of Section 4 and the
stationary Markov equilibrium of Theorem 5.1 can be computed explicitly.
Example 6.1 Suppose that all agents have the same utility function
½
¾
x; x ≤ 1
u(x) =
,
1; x > 1
(6.1)
the same upper-bound on loans k = δ, and the same income distribution
P[Y = 0] = 1 − δ, P[Y = 2] = δ for some 0 < δ < 1/2
(6.2)
so that Q = E(Y ) = 2δ < 1. Suppose also that the bank sets interest rates
1
r1 = , r2 = 1.
δ
(6.3)
We claim then that, for sufficiently small values of the discount parameter, namely β ∈ (0, δ),
and with price
p = 1,
(6.4)
the optimal policy in the one-person game of Section 4 is given as
½
¾
s + δ, 0 ≤ s ≤ 1 − δ
c(s) =
;
1,
s≥1−δ
(6.5)
that the invariant measure µ of the corresponding (optimally controlled) Markov chain (4.12)
has µk ≡ µ({k}) given by
µ−1 = (1 − δ)(1 − η), µ0 = (1 − δ)(1 − η)η, µk = (1 − η)η k
(k ∈ N)
(6.6)
with η , δ/(1 − δ); and that the pair (p, µ) of (6.4) and (6.6) then corresponds to a stationary
Markov equilibrium as in Theorem 5.1.
With c(·) given by (6.5), the amounts borrowed and deposited are given by


½
¾
0 ≤ s ≤ 1 − δ
δ;
0;
0≤s≤1
d(s) = 1 − s; 1 − δ ≤ s ≤ 1
and `(s) =
,
(6.7)
s − 1; s ≥ 1


0;
s≥1
respectively, in the notation of (5.3), whereas the Markov chain of (4.12) takes the form


0 ≤ Sn+ ≤ 1 − δ
−1 + Yn+1 ;
Sn+1 = −(1 − Sn+ )/δ + Yn+1 ; 1 − δ ≤ Sn+ ≤ 1 .

 +
Sn+ ≥ 1
Sn − 1 + Yn+1 ;
After a finite number of steps, this chain only takes values in {−1, 0, 1, 2, ...} with transition
probabilities
p−1,−1 = 1 − δ, p−1,1 = δ;
pn,n+1 = δ, pn,n−1 = 1 − δ
17
(n ∈ N0 ).
The probability measure µ of (6.6) is the unique invariant measure of a Markov Chain with
these transition probabilities.
Consider now the return function Q(s) = I(π)(s), S0 = s corresponding to the stationary
strategy π of (6.5) in the one-person game. This function satisfies Q(s) = u(c(s)) + β · EQ(g(s −
c(s)) + Y ), s ≥ 0 and Q(s) = u(s) + Q(0), s ≤ 0, or equivalently


s + Q(0);
s≤0






(s + δ) + β¡· EQ(−1 ¢+ Y ); 0 ≤ s ≤ 1 − δ
.
(6.8)
Q(s) =
s−1
1 + β · EQ δ + Y ;
1 − δ ≤ s ≤ 1





1 + β · EQ(s − 1 + Y );
s≥1
In order to check the optimality of this strategy for the one-person game, it suffices to show (by
Theorem 4.1) that Q satisfies the Bellman equation Q = T Q (in the notation of (4.7) ). This
verification is carried out in Appendix A of [GKSS].
Let us check now the balance equation (5.2); it takes the form
¾
ZZ ½
Z
d(s+ )
y∧
µ(ds)λ(dy) = d(s+ )µ(ds)
(6.9)
δ
which is satisfied trivially, since both sides are equal to δ(µ({−1}) + µ({0})), from (6.2)–(6.7).
Thus the Assumptions 5.1 and 5.2 are both satisfied, and the pair (p, µ) of (6.4) and (6.6)
corresponds to a stationary Markov equilibrium.
Example 6.2 Suppose that all agents have the same utility function
½
¾
x;
x≤1
u(x) =
1 + η(x − 1); x > 1
(6.10)
for some 0 < η < 1, the same upper-bound on loans k = 1, and the same income distribution
P[Y = 0] = 1 − δ, P[Y = 5] = δ
for some
1
1
<δ< .
3
2
(6.11)
Suppose also that the bank fixes the interest rates r1 = 1δ , r2 = 1 as in (6.3). We claim that
for sufficiently small values of the discount-parameter, namely β ∈ (0, 1/3), for suitable values
of the slope-parameter η ∈ (0, 1) , and with price p = 1 as in (6.4), the optimal policy in the
one-person game of Section 4 is given as
½
¾
1;
0≤s≤2
c(s) =
;
(6.12)
s − 1; s ≥ 2
the invariant measure of the corresponding Markov chain in (4.12) has µk = µ({k}) given by
µ−1/δ = (1 − δ)3 , µ0 = δ(1 − δ)2 , µ1 = δ(1 − δ), µ5−1/δ = δ(1 − δ)2 ,
µ5 = δ 2 (1 − δ), µ6 = δ 2 ;
(6.13)
and the pair (p, µ) of (6.4) and (6.13) corresponds to a stationary Markov equilibrium for the
strategic market game.
18
For the consumption strategy of (6.12), the amounts borrowed and deposited by an agent
with wealth s ≥ 0 are given as d(s) = (1 − s)+ and `(s) = (s − 1) · 1[1,2] (s) + 1(2,∞) (s), respectively, and the Markov Chain of (4.12) becomes
 +

(Sn − 1)/δ + Yn+1 ; 0 ≤ Sn+ ≤ 1
1 ≤ Sn+ ≤ 2 .
Sn+1 = Sn+ − 1 + Yn+1 ;
(6.14)


1 + Yn+1 ;
Sn+ ≥ 2
ª
©
After a finite number of steps, the chain {Sn } takes values in the set − 1δ , 0, 1, 5 − 1δ , 5, 6 with
transition probabilities given by the matrix


1−δ
0
0
δ 0 0
1 − δ
0
0
δ 0 0



 0
1
−
δ
0
0
δ
0
.

(6.15)
 0
0
1 − δ 0 0 δ


 0
0
1 − δ 0 0 δ
0
0
1−δ 0 0 δ
It is not hard to check that the measure µ of (6.13) is the unique invariant measure for a Markov
chain with the transition probability matrix of (6.15).
The optimality of the strategy (6.12) for the one-person game, is verified in Appendix
B of [GKSS]. On the other hand, the balance equation (5.2) takes again the form (6.9) and is
again satisfied trivially, since both sides are now equal to µ−1/δ + µ0 = (1 − δ)2 . Therefore,
Assumptions 5.1 and 5.2 are both satisfied and the pair (p, µ) of (6.4), (6.13) is a stationary
Markov equilibrium, from Theorem 5.1.
Example 6.3 Suppose that all agents have the same utility function
½
¾
2x; x ≤ 0
u(x) =
,
x; x ≥ 0
(6.16)
the same upper bound on loans k = 1, and the same income distribution
1
P[Y = 0] = P[Y = 2] = .
2
(6.17)
In particular, Q = E(Y ) = 1. Suppose also that the bank sets interest rates
r1 = r2 = 2.
(6.18)
We claim then that, with 0 < β < 1/3 and p = 1, the optimal consumption policy is
c(s) = s + 1, s ≥ 0
(6.19)
(borrow up to the limit, and consume everything). The corresponding Markov Chain of (4.12)
becomes trivial, namely Sn+1 = 2(Sn+ − c(Sn+ )) + Yn+1 = −2 + Yn+1 , n ≥ 0 and has invariant
measure µ0 = µ−2 = 1/2. In equilibrium, everybody borrows k = 1, half the agents pay back 2,
the other half pay back nothing, and so the bank balances its books (equation (5.2) is satisfied).
On the other hand, with 1/3 < β < 1/2 and p = 1, we claim that the optimal consumption
policy is
c(s) = s, s ≥ 0,
(6.20)
19
i.e., neither to borrow nor to lend, and to consume everything at hand. The Markov Chain of
(4.12) is again trivial, Sn+1 = Yn+1 , n ≥ 0 and has invariant measure µ0 = µ2 = 1/2; again
the books balance (equation (5.2) is satisfied), because there are neither borrowers nor lenders.
These claims are verified in Appendix C of [GKSS].
7
Two Existence Theorems
From Theorem 5.1 we know that a stationary Markov equilibrium (Definition 3.2) for our strategic market game exists, if (i) each agent’s optimally controlled Markov chain has a stationary
distribution with finite mean, and (ii) the bank balances its books. Condition (i) follows from
natural assumptions about the model, as in Theorems 4.2 and 4.3. However, condition (ii) is
more delicate, and so it is of interest to have existence results that do not rely on this assumption.
We provide two such results in Theorems 7.1 and 7.2 below.
Theorem 7.1 Suppose that the following hold:
(i) Assumption 4.1.
(ii) Agents have common utility function u(·), upper bound k on loans, and income distribution
λ.
R 2
y λ(dy) < ∞.
(iii) λ({0}) = 1 − δ, λ([a, ∞)) = δ for some 0 < δ < 1, 0 < a < ∞;
Then, with interest rates r1 = 1/δ and r2 = 1, the pair (p, µθ ) corresponds to a stationary
Markov equilibrium, for any p ∈ [k/aδ, ∞) and with θ = (r1 , r2 , p).
Proof: Theorem 4.3 guarantees that, under conditions (i) and (iii), the optimally controlled
Markov Chain of (4.12) has an invariant distribution µ = µθ with finite mean, where θ =
(1/δ, 1, p), ∀p ≥ k/aδ. Thus, Assumption 5.1 is satisfied, and in order to prove the result it
suffices (by Theorem 5.1) to check the balance equation (5.2), now in the form
¸
Z
Z
Z ·
1 +
d(s+ )µ(ds).
py ∧ d(s ) µ(ds)λ(dy) =
δ
S
[a,∞) S
(As ρ2 = 0, the bank pays no interest to depositors, and balancing its books means that the
bank gets back from the borrowers exactly what they had received in loans.) Now, for any
s ∈ S, y ≥ a and p ≥ k/aδ, we have py ≥ (k/aδ) · a = (k/δ) ≥ d(s+ )/δ, and thus the left-hand
side of (5.2) equals, by assumption,
Z Z
Z
Z
λ([a, ∞))
1
+
+
d(s )µ(ds)λ(dy) =
d(s )µ(ds) =
d(s+ )µ(ds).
¥
δ
δ
S
S
The conclusion of Theorem 7.1 holds for Examples 6.1 and 6.2; these satisfy its conditions
(ii) and (iii), though not its condition (i). Observe also that, under the conditions of Theorem
7.1, we have
Z
k
Q = yλ(dy) ≥ aλ([a, ∞)) = aδ ≥ .
(7.1)
p
20
Theorem 7.2 Suppose the following hold:
(i), (ii) as in Theorem 7.1.
(iii) λ([0, y ∗ ]) = 1, for some y ∗ ∈ (0, ∞).
(iv) u0 (∞) > 0.
Then there exist interest rates r1 ∈ [1, y ∗ /Q], r2 = 1, a price p ∈ (k/Q, ∞) and a probability
measure µθ on B(S), such that the pair (p, µθ ) with θ = (r1 , r2 , p) corresponds to a stationary
Markov equilibrium.
Note that Example 6.2 satisfies conditions (ii), (iii) and (iv), as well as the conclusion, of
this result.
It seems likely that all of the assumptions of Theorem 7.2 can be weakened. In particular,
it should be possible to replace (ii) by the assumption there are finitely many types of utility
functions and income distributions. A more challenging generalization would be to eliminate
(iv), and perhaps replace (iii) by the assumption that λ has finite second moment.
The rest of this section is devoted to the proof of Theorem 7.2, which will rely on Kakutani’s fixed point theorem. Before applying Kakutani’s theorem, we shall deal with three technical problems: (1) bounding the Markov Chain corresponding to an optimal plan and thereby
bounding the stationary distribution, (2) bounding the price, and (3) finding interest rates to
balance the books.
7.1
Bounding the Markov Chain
Let θ = (r1 , r2 , p) be a vector of parameters for the one-person game of Section 4. The discount
factor will be held constant, but the upper bound on loans will be a function of p, namely
k(p) = pk1 , for some 0 < k1 < Q
(7.2)
where k1 is the bound when the price p is equal to 1. The inequality k(p)/p ≡ k1 < pQ says
that the bank imposes a loan limit strictly less than an agent’s expected income. In order to
guarantee that the books balance, it is intuitively clear that the upper-bound on loans cannot
exceed the monetary value of expected income, as was observed already in (7.1).
To show their dependence on θ, we now write cθ (·) for the optimal consumption function
of Theorem 4.2 as in Remark 4.1 and use Vθ (·) to denote the value function. Likewise, the
function g(·) of (4.2) is written gθ (·) to indicate its dependence on the interest rates r1 and r2 .
We also write Sθ for the state-space [−r1 k(p), ∞).
Let {Sn }n∈N be the Markov chain of successive wealth-levels for an agent who uses cθ (·)
in the one-person game with parameter θ. We can rewrite (4.12) to show the dependence on θ
as
+
+
Sn = gθ (Sn−1
− cθ (Sn−1
)) + pYn , n ≥ 1,
(7.3)
where Y1 , Y2 , ... are independent with common distribution λ. These random variables are uniformly bounded by condition (iii) of Theorem 7.2, so that bounding the chain is tantamount to
bounding the function s 7→ s+ − cθ (s+ ). (The bounding of the price p is treated in the next subsection.) It will also be important to obtain a uniform bound over an appropriate collection of
θ−values. Let us fix p∗ ∈ (0, ∞), r2∗ ∈ [1, 1/β), r1∗ ∈ [r2∗ , ∞) and introduce the parameter-space
Θ , {(r1 , r2 , p) : 1 ≤ r2 ≤ r1 ≤ r1∗ , r2 ≤ r2∗ , 0 < p ≤ p∗ }.
21
(7.4)
Lemma 7.1
η ∗ , sup{|s+ − cθ (s+ )| : θ ∈ Θ, s ∈ Sθ } < ∞.
Proof: Since 0 < cθ (s) ≤ s + k(p) = s + pk1 ≤ s + p∗ k1 , we need only consider those values of
s and θ for which s+ − cθ (s+ ) > 0 and, in particular, s > 0. Furthermore, we have s − cθ (s) =
p[s/p − c(s/p; r1 , r2 , 1)] from (4.13), so that the supremum of s − cθ (s) over Θ is the same as
that over the compact set K , {θ ∈ Θ : p = p∗ }.
Now let s > 0 and s > cθ (s). By Theorem 4.2(d-f) we have that cθ (s) > 0, and that the
number Lθ , lims→∞ (s − cθ (s)) ∈ (0, ∞) satisfies
¶¸
¶
· µ
µ
u0 (∞)
0 cθ (r2 Lθ + pY )
0 cθ (r2 Lθ )
=E u
≤u
.
βr2∗
p
p
With i : (u0 (∞), u0+ (0)) → (0, ∞) denoting the inverse of the function u0 (·) on (0, ∞), we get
then
¶
µ 0
¶
µ 0
u (∞)
u (∞)
∗
cθ (r2 Lθ ) ≤ p · i
≤p ·i
,
r2 β
r2∗ β
since u0 (∞) < u0 (∞)/βr2∗ ≤ u0 (∞)/βr2 < ∞. Define
¶¾
½
µ 0
u (∞)
.
η(θ) , sup s ≥ 0 : cθ (s) ≤ p∗ · i
r2∗ β
(7.5)
Then η(θ) < ∞ for each θ ∈ K because cθ (s) → ∞ as s → ∞ (Theorem 4.2(e)). Also, as
in [KSS2, Proposition 3.4], θ 7→ cθ (s) is continuous for fixed s. This fact, together with the
continuity and monotonicity of cθ (·), can be used to check that η(·) is upper-semicontinuous.
Therefore, for θ ∈ K and s > cθ (s), s > 0, we have s − cθ (s) ≤ r2 (s − cθ (s)) ≤ r2 Lθ ≤
supK η(θ) < ∞.
¥
Lemma 7.2 Let {Sn }n∈N be the Markov chain (7.3) corresponding to optimal play in the oneperson game with parameter θ ∈ Θ. Then, whatever the distribution of S0 , the distributions of
Sn+ , n ≥ 1, are supported on the interval [0, r1∗ η ∗ + p∗ y ∗ ].
Proof: Immediate from (7.2), (7.3), and Lemma 7.1.
7.2
¤
Bounding the Price
Assume that the total amount of fiat money in our many-person model is the positive quantity
W . (Recall from Lemma 3.1 that W is preserved from period to period.) Let γ be the distribution
of fiat money among agents. Notice that γ differs from the distribution of wealth positions µ,
in that those agents with negative wealth-positions hold no fiat money. Thus µ(A) = γ(A ∩
[0, ∞)) for A ∈ B(R).
Suppose that in a certain period the parameters of the model are given by the vector
θ = (r1 , r2 , p) and that all agents bid according to cθ (·). The newly formed price is
Z
Z
1
1
p̃ = p̃(θ, γ) ,
cθ (s)γ(ds) =
cθ (s+ )µ(ds).
(7.6)
Q [0,∞]
Q R
Let Θ be as in (7.4) with
p∗ ,
W∗
Q − k1
and r1∗ ,
22
y∗
,
Q
(7.7)
where W ∗ is a given constant, 0 < W ∗ ≤ W. (Recall that 0 < k1 < Q and y ∗ is an upper bound
on the income variable Y .)
Define M to be the collection of all probability measures γ on B([0, r1∗ η ∗ + p∗ y ∗ ]) with
R
∗
(0,∞) sγ(ds) = W . We need a technical lemma.
Lemma 7.3
(a) Suppose θn → θ as n → ∞, where θ, θ1 , θ2 , ... lie in Θ. Then cθn (s) → cθ (s) uniformly on
compact sets.
(b) The function p̃(θ, γ) of (7.6) is continuous and everywhere positive on Θ×M. Furthermore,
p̃ has a continuous, everywhere positive extension to the compact set Θ̄ × M, where
Θ̄ = {(r1 , r2 , p) : 1 ≤ r2 ≤ r1 ≤ r1∗ , r2 ≤ r2∗ , 0 ≤ p ≤ p∗ }.
Proof: (a) Similar to Proposition 3.4 of [KSS2].
(b) The continuity of p̃ on Θ × M follows from (a), since every γ ∈ M is supported by the
compact set K , [0, r1∗ η ∗ + p∗ y ∗ ]. Also, p̃ is strictly positive on Θ × M because, by Theorem
4.2(d), cθ (s) > 0 for all s > 0. To extend to Θ̄ × M, let θ = (r1 , r2 , 0) ∈ Θ̄, and first set
cθ (s) ≡ c(s; r1 , r2 , 0) , s. Then, for γ ∈ M, let
Z
1
W∗
.
p̃(θ, γ) ,
sγ(ds) =
Q (0,∞)
Q
Obviously our extension is positive. To check its continuity, fix θ = (r1 , r2 , 0) ∈ Θ̄, γ ∈ M, and
suppose that limn→∞ (θn , γn ) = (θ, γ), where (θn , γn ) ∈ Θ × M for all n. It suffices to show that
Z
Z
1
1
W∗
, as n → ∞.
p̃(θn , γn ) =
cθn (s)γn (ds) →
sγ(ds) =
Q
Q
Q
(n)
(n)
(n)
(n)
Suppose θn = (r1 , r2 , pn ). By (7.4) and Lemma 7.1, we have |cθn (s)−s| = pn |c(s/pn ; r1 , r2 , 1)−
s/pn | ≤ pn η ∗ → 0, as n → ∞, and the result follows.
¥
Define
p∗ , inf{p̃(θ, γ) : (θ, γ) ∈ Θ × M}.
Lemma 7.4 For every (θ, γ) ∈ Θ × M, we have
0 < p∗ ≤ p̃(θ, γ) ≤ p∗ =
(7.8)
W∗
Q−k1 .
Proof: The first inequality is by Lemma
third, use
R 7.3(b), and the
R second by (7.8). ∗For the
∗
(7.2), (7.6), and (7.7) to get Qp̃(θ, γ) = cθ (s)γ(ds) ≤ (s + pk1 )γ(ds) ≤ W + p k1 = p∗ Q. ¥
7.3
Finding Interest Rates That Balance the Books
Let the sets Θ and M be as in the previous section so that, in particular, p∗ and r1∗ satisfy (7.7).
Suppose that γ ∈ M is the distribution of money across agents at some stage of play. Assume
also that all agents believe a certain θ = (r1 , r2 , p) ∈ Θ to be the vector of parameter-values. If
they further believe the game to be in equilibrium, then they will play according to cθ (·). Our
objective in this section is to see that in such a situation the bank can find new interest rates
23
(r̃1 , r̃2 ) that will balance the books. To do so, we need expressions for the total amounts of fiat
money borrowed and paid back.
An agent with money s ≥ 0 will borrow the amount dθ (s) = (cθ (s) − s)+ , so the total
amount borrowed is
Z
D ≡ D(θ, γ) , dθ (s)γ(ds).
(7.9)
If the bank sets the interest rate r̃1 for borrowers, then the amount paid back by an agent, who
begins the period with fiat money s and receives income p̃y, is p̃y ∧ r̃1 dθ (s), where p̃ is the newly
formed price as in (7.6). The total amount paid back is R(r̃1 ), where
ZZ
R(r) ≡ R(r, θ, γ) ,
{p̃y ∧ rdθ (s)}γ(ds)λ(dy)
(7.10)
and λ is the distribution of the generic income variable Y . Let
J ≡ J(θ, γ) , {r ∈ [1, r1∗ ] : R(r, θ, γ) = D(θ, γ)}.
(7.11)
Lemma 7.5 For all θ ∈ Θ and γ ∈ M, the set J(θ, γ) is a closed, nonempty subinterval of
[1, r1∗ ]. In particular, there exists r ∈ [1, r1∗ ] such that R(r, θ, γ) = D(θ, γ).
Proof: The function r 7→ R(r) = R(r, θ, γ) of (7.10) is obviously nondecreasing, and is continuous by the dominated convergence theorem; thus J is clearly a closed subinterval of [1, r1∗ ]. It
remains to prove that J is nonempty, and for this it suffices to show that
R(1) ≤ D ≤ R(r1∗ ).
The first inequality is trivial, because p̃y ∧ 1dθ (s) ≤ dθ (s). To prove the second inequality, let
c∗ = cθ (0). By Theorem 4.2(b), we have cθ (s) ≥ cθ (0) = c∗ and dθ (s) ≤ dθ (0) = c∗ , s ≥ 0.
Thus, (7.6) implies Qp̃ ≥ c∗ . Fix s ≥ 0, y ∈ [0, y ∗ ] and observe
p̃y ∧ r1∗ dθ (s) ≥
Hence,
R(r1∗ )
1
≥
Q
ZZ
y∗
y
y
c∗
y ∧ dθ (s) ≥ [c∗ ∧ dθ (s)] = dθ (s).
Q
Q
Q
Q
D
{dθ (s)y}γ(ds)λ(dy) =
·
Q
Z
yλ(dy) = D.
¤
Remark 7.1 By Lemma 7.5, we see that the bank can choose r̃1 in a given period so that
borrowers, as a group, pay back precisely the amount that they borrowed. The bank can then
set r̃2 = 1, which means that depositors, as a group, get back exactly what they received: the
bank is able to balance its books. It would be interesting to have conditions that make it possible
for the bank to pay a positive interest rate ρ2 > 0 (r̃2 ≡ 1 + ρ2 > 1) to depositors, and still
balance its books. It seems unlikely that this will always be possible without price inflation, or
growth in the economy.
24
7.4
The Proof of Theorem 7.2
Consider the set
Θ̃ = {(r1 , r2 , p) : 1 ≤ r1 ≤ r1∗ , r2 = 1, p∗ ≤ p ≤ p∗ }
where p∗ , r1∗ and p∗ are given by (7.7)R and (7.8), respectively. Let M be the set of probability
measures γ on B([0, r1∗ η ∗ +p∗ y ∗ ]) with sγ(ds) = W ∗ , as in the previous two subsections. Define
a set-valued mapping ψ on the compact set Θ̃ × M as follows: for (θ, γ) = ((r1 , r2 , p), γ) ∈
Θ̃ × M, ψ((r1 , r2 , p), γ) is the set of all (θ̃, γ̃) = ((r̃1 , r̃2 , p̃), γ̃) such that r̃1 ∈ J(θ, γ), r̃2 = 1,
p̃ = p̃(θ, γ), and γ̃ is the distribution of [gθ (S0+ −cθ (S0+ ))+ p̃Y ]+ . Here S0+ and Y are independent.
By Lemmata 7.2, 7.4, and 7.5, ψ(θ, γ) is a compact, convex subset of Θ̃ × M. Furthermore, it
is straightforward to verify that
{((θ, γ), (θ̃, γ̃)) : (θ̃, γ̃) ∈ ψ(θ, γ)} is a closed subset of
(Θ̃ × M) × (Θ̃ × M).Hence,ψ is upper semicontinuous (cf. Theorem 10.2.4 of Istrătescu (1981)),
and thus, by Kakutani’s fixed point theorem (Corollary 10.3.10 in Istrătescu (1981)), there exists
(θ, γ) = ((r1 , r2 , p), γ) ∈ Θ̃ × M such that (θ, γ) ∈ ψ(θ, γ).
This fixed point ((r1 , r2 , p), γ) = (θ, γ) determines a stationary Markov equilibrium in
which the bank sets the interest rates to be r1 and r2 = 1, the price is p, and the distribution of
wealth-levels µ is related to γ, the distribution of fiat money, by the rule that µ is the distribution
of gθ (S0+ − cθ (S0+ )) + pY ; here S0+ and Y are independent. Clearly, we have 1 = r2 ≤ r1 ≤ r1∗ =
y∗ /Q, and k = k(p) = pk1 < pQ, from (7.2). ¤
8
The Game with a Money-Market
We shall discuss in this section the strategic market game when there is no outside bank, but
instead agents can borrow or deposit money through a money-market at interest rates r1 and
r2 , respectively, with r1 > r2 . In contrast to the situation of an outside bank, which fixes and
announces interest rates for borrowing and depositing money, here r1 and r2 are going to be
determined endogenously.
In order to see how this can be done, imagine that agent α ∈ I enters the day t = n with
α (w) from the previous day — in particular, with fiat money (S α (w))+ .
wealth-position Sn−1
n−1
α
His information Fn−1
(at the beginning of day t = n − 1) measures, in addition to the quantities
mentioned in Section 3, past interest rates r1,k and r2,k , k = 0, ..., n − 1 for borrowing and
depositing, respectively. The agent can decide either to deposit money
α
`αn (w) ∈ [0, (Sn−1
(w))+ ]
(8.1)
into the money-market, or to offer a bid of
jnα (w) ∈ [0, k α ]
(8.2)
in I.O.U. notes for money, or to do neither, but not both:
jnα (w) · `αn (w) = 0.
The total amount deposited is
(8.3)
Z
Ln (w) ,
I
`αn (w)ϕ(dα);
25
(8.4)
the total amount offered in I.O.U. notes is
Z
Jn (w) ,
I
jnα (w)ϕ(dα);
(8.5)
and the money-market is declared active on day t = n, if
Jn (w) · Ln (w) > 0
(8.6)
(inactive, if Jn (w) · Ln (w) = 0).
After agents have thus made their bids in the money-market, a new interest rate for
borrowing money is formed, namely
)
(
Jn (w)
;
if
J
(w)
·
L
(w)
>
0
n
n
.
(8.7)
r1,n (w) , Ln (w)
1;
otherwise
Agent α ∈ I receives his I.O.U. notes’ worth jnα (w)/r1,n (w) in fiat money, and bids the amount
( α
)
jn (w)
α (w); if J (w) · L (w) > 0
−
`
n
n
α
α
+
n
bn (w) , (Sn−1 (w)) + r1,n (w)
(8.8)
0;
otherwise
in the commodity market. Thus, the total amount of money bid for commodity is
(
)
Z
Jn (w)
α (w); if J (w) · L (w) > 0
−
L
n
n
α
n
r
(w)
Bn (w) ,
bn (w)ϕ(dα) = Wn−1 (w) + 1,n
0;
otherwise
I
= Wn−1 (w)
(8.9)
from (8.8), where
Z
Wk (w) ,
I
(Skα (w))+ ϕ(dα)
(8.10)
is the total amount of money across agents on day t = k ∈ N0 .
Next, the various agents’ commodity endowments Ynα (w), α ∈ I for that day t = n are
revealed (same assumptions and notation as in the beginning of Section 3). A new commodity
price
Wn−1 (w)
Bn (w)
=
(8.11)
pn (w) ,
Q
Q
is formed, and agent α ∈ I receives his bid’s worth xαn (w) , bαn (w)/pn (w) in units of commodity.
He consumes this amount, and derives utility ξnα (w) as in (3.4). The borrowers pay back their
debts – with interest r1,n (w) – to the extent that they can; the rest is forgiven, but “punishment
in the form of negative-utility” is incurred if they enter the next day with Snα (w) < 0, as in (3.4).
A new interest rate for deposits is formed
¾
½ 1 R α
{j (w) ∧ pn (w)Ynα (w)}ϕ(dα); if Jn (w) · Ln (w) > 0
I
L
(w)
n
(8.12)
r2,n (w) ,
1;
otherwise
and agent α ∈ I moves to the new wealth-position
Snα (w) , [r2,n (w)`αn (w) − jnα (w)] · 1{Jn (w)·Ln (w)>0} + pn (w)Ynα (w)
α
= g((Sn−1
(w))+ − bαn (w);r1,n (w),r2,n (w)) + pn (w)Ynα (w)
in the notation of (4.2).
26
(8.13)
Remark 8.1 Indeed, suppose that the money-market is active on day t = n. If agent α ∈ I
α (w))+ − `α (w) <
is a depositor (`αn (w) > 0, jnα (w) = 0), he bids the amount bαn (w) = (Sn−1
n
α
+
(Sn−1 (w)) in the commodity market, and ends up with fiat money
α
Snα (w) = r2,n (w) · [(Sn−1
(w))+ − bαn (w)] + pn (w)Ynα (w),
after he has received his endowment’s worth and his deposit back with interest. If agent α is
a borrower (jnα (w) > 0, `αn (w) = 0), he bids in the commodity market the amount bαn (w) =
α (w))+ + [1/r
α
α
+
(Sn−1
1,n (w)]jn (w) > (Sn−1 (w)) , and his new wealth-position is
α
Snα (w) = −jnα (w) + pn (w)Ynα (w) = r1,n (w) · [(Sn−1
(w))+ − bαn (w)] + pn (w)Ynα (w).
If the agent is neither borrower nor depositor (or if the money-market is inactive) on day
α (w))+ for commodity and ends up at the new wealth-position
t = n, he bids bαn (w) = (Sn−1
α
α
Sn (w) = pn (w)Yn (w).
Remark 8.2 These rules preserve the total amount of fiat money in the economy, and guarantee
that the price of the commodity remains constant from period to Rperiod. Indeed, if the moneymarket is inactive on day t = n, we have
Wn (w) = pn (w) I Ynα (w)ϕ(dα) = Qpn (w) =
Wn−1 (w) in the notation of (8.11), from (3.14) and (8.10) . On the other hand, if the moneymarket is active on day t = n , we obtain from (8.13) and (8.12):
Z
Z
α
+
Wn (w) = (Sn (w)) ϕ(dα) = [r2,n (w)`αn (w) + pn (w)Ynα (w)1{jnα (w)=0} ]ϕ(dα)
I
I
Z
+
[pn (w)Ynα (w) − jnα (w)]1{0<jnα (w)≤pn (w)Ynα (w)} ϕ(dα)
I
Z
= r2,n (w)Ln (w) + pn (w)Ynα (w)1{jnα (w)≤pn (w)Ynα (w)} ϕ(dα)
I
Z
α
−
jn (w)1{jnα (w)≤pn (w)Ynα (w)} ϕ(dα)
I
Z
= ([jnα (w)∧pn (w)Ynα (w)]+[pn (w)Ynα (w)−jnα (w)]1{jnα (w)≤pn (w)Ynα (w)} )ϕ(dα)
I
Z
= pn (w) Ynα (w)ϕ(dα) = Qpn (w) = Wn−1 (w),
I
again. In either case,
Wn = W0 =: W, pn = p0 ,
W
, ∀n ∈ N.
Q
(8.14)
α −measurable
Definition 8.1 A strategy π α for agent α ∈ I specifies w 7→ `αn (w), w 7→ jnα (w) as Fn−1
(thus also Fn−1 − random variables that satisfy (8.1) and (8.2) for every n ∈ N.
A strategy π α is called stationary, if it is of the form
α
jnα (w) = j α ((Sn−1
(w))+ ; pn−1 (w), r1,n−1 (w), r2,n−1 (w))
α
`αn (w) = `α ((Sn−1
(w))+ ; pn−1 (w), r1,n−1 (w), r2,n−1 (w))
∀ n ∈ N; here j α , `α are measurable mappings of [0, ∞) × Θ̌ into R with
0 ≤ j α (s; θ) ≤ k α , 0 ≤ `α (s; θ) ≤ s, j α (s; θ) · `α (s; θ) = 0;
27
∀(s, θ) ∈ [0, ∞) × Θ̌
(8.15)
Θ̌ , {(r1 , r2 , p) : 1 ≤ r2 ≤ r1 < ∞, r2 < 1/β, p > 0} .
(8.16)
(Such a strategy requires, of course, the specification of an initial vector of interest rates and
price θ0 = (r1,0 , r2,0 , p0 ) ∈ Θ̌, in order for j1α , `α1 to be well-defined.)
A collection of strategies Π = {π α : α ∈ I} is called admissible for the money-market
game if, for every n ∈ N , the functions (α, w) 7→ `αn (w), (α, w) 7→ jnα (w) are Gn−1 ≡
B(I) ⊗ Fn−1 −measurable, in the notation of subsection 3.1.
Definition 8.2 We say that an admissible collection of stationary strategies Π̃ = {π̃ α : α ∈ I}
results in stationary Markov equilibrium (r1 , r2 , p, µ) for the money-market game, with θ =
(r1 , r2 , p) ∈ Θ̌ and µ a probability measure on B(S)if, starting with initial vector (r1,0 , r2,0 , p0 ) =
θ, and with ν0 = µ in the notation of (3.13), we have
(i) (r1,n , r2,n , pn ) = θ, νn = µ (∀n ∈ N) when agents play according to the strategies π̃ α ,
α ∈ I, and
(ii) as in Definition 3.1.
¥
In an effort to seek sufficient conditions for such a stationary Markov equilibrium, let us
assume from now on that all agents have the same utility function uα (·) ≡ u(·), the same upper
bound on loans k α ≡ k, and the same income distribution λα ≡ λ. By analogy with Assumptions
5.1 and 5.2, consider now the following:
Assumption 8.1 Suppose that there exists a triple θ = (r1 , r2 , p) ∈ Θ̌ for which the one-person
game of Section 4
(i) has a unique optimal stationary plan π, corresponding to a continuous consumption function c ≡ cθ : [0, ∞) → [0, ∞), and j(s; θ) = r1 (cθ (s) − s)+ ≡ r1 d(s), `(s; θ) ≡ `(s) =
(s − cθ (s))+ as in (8.14), (5.3); and
(ii) the associated RMarkov Chain of wealth-levels in (4.10) has an invariant measure µ ≡ µθ
on B(S) with sµ(ds) < ∞.
Assumption 8.2 The quantities of Assumption 8.1 satisfy the balance equations
Z
Z
d(s+ )µ(ds) = `(s+ )µ(ds) > 0
(8.17)
(“total amount borrowed is positive, and equals total amount deposited, in equilibrium”) and
Z
ZZ
r2 `(s+ )µ(ds) =
[r1 d(s+ ) ∧ py]µ(ds)λ(dy)
(8.18)
(“total amount paid back to depositors equals total amount paid back by borrowers, in equilibrium”).
The reader should not fail to notice that we have now two balance equations (8.16) and
(8.17), instead of the single balance equation (5.2) for the outside bank. This reflects the fact
that the bank needs to balance its books only once, whereas a money-market has to clear twice:
28
(i) before the agents’ endowments are announced — by the formation of the “ex ante”
interest rate (8.7), which guarantees that the deposits Ln (w) exactly match the payments to borrowers Jn (w)/r1,n (w),
(ii) and after — by the Rformation of the “ex post” interest rate (8.12), which matches
exactly the amount I [jnα (w) ∧ pn (w)Ynα (w)]ϕ(dα) paid back to borrowers, with the
amount r2,n (w)Ln (w) that has to be paid to depositors.
In light of these remarks, it is no wonder that stationary Markov equilibrium with a
money-market is much more delicate, and difficult to construct, than with an outside bank.
This extra difficulty will also be reflected in the Examples that follow.
Here are now the analogues of Lemma 5.1 and Theorem 5.1; their proofs are left as an
exercise for the diligent reader.
Lemma 8.1 Under Assumptions 8.1 and 8.2,
Z
Z
1
1
+
c(s )µ(ds) =
s+ µ(ds).
p=
Q
Q
(8.19)
Theorem 8.1 Under Assumptions 8.1 and 8.2, the family Π = {π α : α ∈ I} with π α = π
(∀α ∈ I) results in Stationary Markov Equilibrium (r1 , r2 , pn , µ) for a money-market.
For fixed θ = (r1 , r2 , p) ∈ Θ̌, Theorems 4.2 and 4.3 provide fairly general sufficient conditions that guarantee the validity of Assumption 8.1. However, we have not been able to obtain
results comparable to Theorems 7.1 and 7.2, providing reasonably general sufficient conditions
for Assumption 8.2 to hold. We shall leave this subject to further research, but revisit in our
new context the Examples of Section 6.
Example 6.1 (continued): Recall the setup of (6.1)–(6.4), the consumption strategy c(·)
of (6.5), and the invariant measure µ of (6.6). The balance equation (5.2), for an outsidebank stationary Markov equilibrium, was satisfied for all values of the Bernoulli parameter
0 < δ < 1/2; however, the balance equations (8.16), (8.17) are satisfied if and only if δ = 1/4.
Thus, for this value δ = 1/4, the vector θ = (r1 , r2 , p) = (4, 1, 1) and the measure
µ({−1}) = 1/2, µ({0}) = 1/6, µ({k}) = 2/3k+1 , k ∈ N of (6.6), form a stationary Markov
equilibrium for the money-market.
Example 6.2 (continued): Recall the setup of (6.11)–(6.14), the consumption function of
(6.13), and the invariant probability measure of (6.14). The balance equation (5.2) for an outside
bank holds for all values of the Bernoulli parameter δ ∈ (1/3, 1/2) = (.33, .5) and all values of
the discount and slope parameters β, η as in (B.10). The balance equations for a money-market
(8.16),
(8.17) will be satisfied, if and only if the total amount Rborrowed in equilibrium, namely
R
+
d(s )µ(ds) = µ−1/δ +µ0 = (1−δ)2 , equals the total amount `(s+ )µ(ds) = µ5−1/δ +µ5 +µ6 =
√
δ deposited in equilibrium; in other words, if and only if δ = (3 − 5)/2 = .382. With this value
of δ in (6.12) and (6.15), the vector θ = (r1 , r2 , p) = (2.62, 1, 1) and the measure µ of (6.14) form
a stationary Markov equilibrium for the money-market.
Example 6.3 (continued): In the setting of (6.16)–(6.18) and with θ = (r1 , r2 , p) = (2, 2, 1),
µ({0}) = µ({2}) = 1/2, the pair (θ, µ) leads to stationary Markov equilibrium if 1/3 < β < 1/2;
no such equilibrium exists for θ = (2, 2, 1) and 0 < β < 1/3.
29
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J. Geanakoplos
Yale University
Box 208281
New Haven, CT 06520–8281
(203) 432–3397
[email protected]
I. Karatzas
Columbia University
619 Mathematics
New York, NY 10027
(212) 854–3177
[email protected]
M. Shubik
Yale University
Box 208281
New Haven, CT 06520–8281
(203) 432–3704
[email protected]
W. Sudderth
University of Minnesota
School of Statistics
Minneapolis, MN 55455
(612) 625–2801
[email protected]
30